Optimal Consumption and Portfolio Decisions With Partially Observed Real Prices

Abstract

We consider optimal consumption and portfolio investment problems of an investor who is interested in maximizing his utilities from consumption and terminal wealth subject to a random inflation in the consumption basket price over time. We consider two cases: (i) when the investor observes the basket price and (ii) when he receives only noisy observations on the basket price. We derive the optimal policies and show that a modified Mutual Fund Theorem consisting of three funds holds in both cases. The compositions of the funds in the two cases are the same, but in general the investor's allocations of his wealth into these funds will differ. However, in the particular case when the investor has constant relative risk-aversion (CRRA) utility, his optimal investment allocations into these funds are also the same in both cases.

Portfolio management with stochastic interest rates and inflation ambiguity

“…Although ρ SZ and ρ Zβ were not estimated by Brennan and Xia (2002), we perform the analysis for their values equal to 0.3 and −0.3. These values of ρ SZ were used in Bensoussan, Keppo, and Sethi (2009). Estimations of Fama and Schwert (1977), Gultekin (1983), Ferson and Harvey (1991), and Moerman and van Dijk (2010) also show that these correlation coefficients can be quite different.…”

“…We assume that the correlation coefficients take values in the interval (−1, 1). The process (2.4) and Ornstein-Uhlenbeck process (2.5) are quite common in modeling the price level and the expected inflation rate, respectively, see for example, Brennan and Xia (2002) and Bensoussan, Keppo, and Sethi (2009). However, in Brennan and Xia (2002) the expected inflation rate is assumed to be observable, whereas in Bensoussan, Keppo, and Sethi (2009) the expected inflation rate is constant, but the price level is assumed to be unobservable.…”

A filtering problem with uncertainty in observation

“…Based on the filtering technique, stochastic optimal control problems with partial information (or observation) have been studied extensively. In the field of finance and insurance, for example, Bensoussan and Keppo [3] and Lakner [15] considered the optimal consumption and portfolio investment problems of an investor who is interested in maximizing his utilities from consumption and terminal wealth under partial information; Xiong and Zhou [24] considered the mean-variance portfolio selection problems under partial information. In the field of stochastic control, Duncan and Pasik-Dunan [5] and [6] considered respectively the optimal control for a partially observed linear stochastic system with an exponential quadratic cost and with fractional brownian motions; Tang [21] gave the maximum principle for partially observed optimal control problems of stochastic differential equations; Wang and Wu [23] studied the Kalman-Bucy filtering equation of a certain forwardbackward stochastic differential equation system and solved a partially observed linear quadratic optimal control problem, and so on.…”

Continuous-time mean-variance asset-liability management with stochastic interest rates and inflation risks

“…Siu [23] added the impact of (macro)economic conditions to the settings in Zhang [30]. Some other works on maximizing the CRRA utility function include Brennan and Xia [2], Munk et al [19] for one risky asset and Bensoussan [1] for multiple risky assets and partial information. Yao et al [27] studied a Markowitz mean-variance defined contribution pension fund management with inflation risk, and obtained closed-form solutions of the efficient strategy and efficient frontier by using the dynamic programming approach.…”